In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a disk (also
spelled disc)
[.] is the region in a
plane bounded by a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not.
For a radius,
, an open disk is usually denoted as
and a closed disk is
. However in the field of
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
the closed disk is usually denoted as
while the open disk is
.
Formulas
In
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, the ''open disk'' of center
and radius ''R'' is given by the formula
:
while the ''closed disk'' of the same center and radius is given by
:
The
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
of a closed or open disk of radius ''R'' is π''R''
2 (see
area of a disk).
Properties
The disk has
circular symmetry.
The open disk and the closed disk are not topologically equivalent (that is, they are not
homeomorphic), as they have different topological properties from each other. For instance, every closed disk is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
whereas every open disk is not compact. However from the viewpoint of
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
they share many properties: both of them are
contractible
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
and so are
homotopy equivalent to a single point. This implies that their
fundamental groups are trivial, and all
homology groups are trivial except the 0th one, which is isomorphic to Z. The
Euler characteristic of a point (and therefore also that of a closed or open disk) is 1.
Every
continuous map from the closed disk to itself has at least one
fixed point (we don't require the map to be
bijective or even
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
); this is the case ''n''=2 of the
Brouwer fixed point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simple ...
. The statement is false for the open disk:
Consider for example the function
which maps every point of the open unit disk to another point on the open unit disk to the right of the given one. But for the closed unit disk it fixes every point on the half circle
As a statistical distribution
A uniform distribution on a unit circular disk is occasionally encountered in statistics. It most commonly occurs in operations research in the mathematics of urban planning, where it may be used to model a population within a city. Other uses may take advantage of the fact that it is a distribution for which it is easy to compute the probability that a given set of linear inequalities will be satisfied. (
Gaussian distributions in the plane require
numerical quadrature.)
"An ingenious argument via elementary functions" shows the mean
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore ...
between two points in the disk to be ,
[J. S. Lew et al., "On the Average Distances in a Circular Disc" (1977).] while direct integration in polar coordinates shows the mean squared distance to be .
If we are given an arbitrary location at a distance from the center of the disk, it is also of interest to determine the average distance from points in the distribution to this location and the average square of such distances. The latter value can be computed directly as .
Average distance to an arbitrary internal point
To find we need to look separately at the cases in which the location is internal or external, i.e. in which , and we find that in both cases the result can only be expressed in terms of
complete elliptic integrals.
If we consider an internal location, our aim (looking at the diagram) is to compute the expected value of under a distribution whose density is for , integrating in polar coordinates centered on the fixed location for which the area of a cell is ; hence
Here can be found in terms of and using the
Law of cosines. The steps needed to evaluate the integral, together with several references, will be found in the paper by Lew et al.;
[ the result is that
where and are complete elliptic integrals of the first and second kinds. ; .
]
Average distance to an arbitrary external point
Turning to an external location, we can set up the integral in a similar way, this time obtaining
where the law of cosines tells us that and are the roots for of the equation
Hence
We may substitute to get
using standard integrals.
Hence again , while also[Abramowitz and Stegun, 17.3.11 et seq.]
See also
* Unit disk, a disk with radius one
*Annulus (mathematics)
In mathematics, an annulus (plural annuli or annuluses) is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word ''anulus'' or ''annulus'' meani ...
, the region between two concentric circles
* Ball (mathematics), the usual term for the 3-dimensional analogue of a disk
* Disk algebra, a space of functions on a disk
* Disk segment
*Orthocentroidal disk
In geometry, the orthocentroidal circle of a non-equilateral triangle is the circle that has the triangle's orthocenter and centroid at opposite ends of its diameter. This diameter also contains the triangle's nine-point center and is a sub ...
, containing certain centers of a triangle
References
{{DEFAULTSORT:Disk (Mathematics)
Euclidean geometry
Circles
Planar surfaces